Electric Field Regression and Error Variance Estimation for Transcranial Magnetic Stimulation using Deep Neural Networks

Transcranial magnetic stimulation (TMS) is a widely used non-invasive neurostimulation technique in neuroscience and in the treatment of psychiatric disorders. By placing a TMS coil over a patient's head, neurons in the brain can be electromagnetically stimulated through the induction of an electric field (E-field). Accurate estimation of the E-field induced in a patient's head is crucial for determining the stimulated area of the brain. The electromagnetic simulation for E-field estimation involves two processes: the development of a volume conductor model (VCM) to determine the electrical conductivity at each position of the brain from a head magnetic resonance (MR) image, and the computation of the E-field on the VCM. Currently, neither of these processes can be performed in real-time. Achieving real-time estimation would greatly assist in determining the appropriate coil position and direction to stimulate the target regions in the patient's brain. In recent years, several methods utilizing deep neural networks (DNNs) have been proposed to estimate E-fields from MR images in real-time. These methods construct a regressor of the E-field using a set of simulated E-fields as training data to estimate the E-field. However, the reliability of these regressors in clinical applications could be improved by incorporating uncertainty estimation of the regressed variables, although this has not been reported. In this study, we enhanced the accuracy of E-field strength estimation by first regressing the E-field and then computing the norm of the E-field vectors, instead of directly regressing the E-field strength. In addition, we investigated the statistical uncertainty of the regressed E-fields using DNN. It should be noted that the E-fields estimated by the regressors are random variables. To evaluate the uncertainty of this application, we employed MCDropout, a well-known Bayesian estimation method. The uncertainty of the regressed E-field was evaluated for each anatomical tissue of the brain, to examine the relationship between uncertainty and depth from the coil. The experimental results of this evaluation are presented quantitatively.